I'm curious what the differences are, in terms of contributing to violent societies and communities, among poverty, relative poverty, and lack of upward mobility.
I think a better way to get at what you're looking to find out might be to reframe your question.
1) Due to the population size and variability of laws by state, county, and municipality, it's difficult to get a national perspective on the effect of gun legislation. You'd likely have to isolate a specific geographic area that had a specific change in gun laws to see what if any effects there were.
2) You'd have to specify what you're going to measure, e.g. reports of shots fired, aggravated assault, homicides, etc. What might be best is to look at the ratio of gun related assaults to homicides over a long period of time, usually 5-10 years before and after.
3) Once you have your population and measurement set up, you'll need to account for any other variables like violent crime rate and economic circumstances in that 10-20 year period and also factor in what the implementation process of the law looked like.
At present, gun restrictions are tepid at best because of the 2nd amendment, so you also have to factor in the limits of what "restrictive" even means. There's little comparison to the UK or Australia because 2A means there are just many more guns in circulation in the US in general. Also, due to court challenges and changing legislative circumstances, gun laws are dynamic. You'd have to find a specific law that has been relatively unchanged in it's its lifespan and geographic area over your 10-20 year look. Even then, you'd maybe come up with a correlation between the law and the circumstances on the ground, but it would be very difficult to say one caused the other.
There are cruder ways to get at an answer, mostly having to do with uniform crime reporting statistics controlled on a per capita basis, but again, you have to make sure the law stays constant. A lot of gun lobbyists like to invoke Chicago, for example, but both the state of Illinois and the city of Chicago have significantly relaxed gun laws over the past decade. There are also federal court findings that can easily sweep away provisions of gun laws Nationwide.
I come from a city where gun possession sent about a quarter of the population either to prison or the morgue, but luckily my mom got me out of there young. What I found is that stricter gun laws just sent more young low income black men to prison, retrenching them in crime and making them more likely to want or need a gun if and when they got out. Now they have vigils every time there's a shooting and parties everytime the city goes 30 days without a shooting. Some of it's working, but so long as there are scores to settle, blocks to hold down, and product to protect, guns will be part of the culture.